NIMCET 2023 — Mathematics PYQ

1. Understand Given Conditions

• $A\cap X=\varnothing ⟹$ Sets $A$ and $X$ share no common elements (disjoint).

• $B\cap X=\varnothing ⟹$ Sets $B$ and $X$ share no common elements (disjoint).

• $A\cup X=B\cup X⟹$ The total combined elements of $A$ and $X$ are exactly identical to the combined elements of $B$ and $X$.

2. Mathematical Proof Using Distributive Law

We are given:

$$A\cup X=B\cup X$$

Take the intersection with set $A$ on both sides:

$$A\cap (A\cup X)=A\cap (B\cup X)$$

• Left-Hand Side (LHS): By absorption law, $A\cap (A\cup X)=A$.

• Right-Hand Side (RHS): Apply the distributive law:

  $$A\cap (B\cup X)=(A\cap B)\cup (A\cap X)$$

Since we know $A\cap X=\varnothing$, substitute it into the expression:

$$A=(A\cap B)\cup \varnothing$$

$$A=A\cap B\text{— (Equation 1)}$$

Equation 1 implies that $A$ is completely contained within $B$ ($A\subseteq B$).

3. Repeat the Process for Set $B$

Take the original union equality again:

$$A\cup X=B\cup X$$

Now, take the intersection with set $B$ on both sides:

$$(A\cup X)\cap B=(B\cup X)\cap B$$

• Right-Hand Side (RHS): By absorption law, $(B\cup X)\cap B=B$.

• Left-Hand Side (LHS): Apply the distributive law:

  $$(A\cap B)\cup (X\cap B)=B$$

Since we know $B\cap X=\varnothing ⟹X\cap B=\varnothing$, substitute it into the expression:

$$(A\cap B)\cup \varnothing =B$$

$$A\cap B=B\text{— (Equation 2)}$$

Equation 2 implies that $B$ is completely contained within $A$ ($B\subseteq A$).

4. Conclusion

From Equation 1 and Equation 2:

$$A=A\cap B=B$$

Therefore, the two sets must be completely identical:

$$A=B$$